Bifurcation Theory and Abrupt Climate Change

在数学方程系统的研究中，分叉理论处理的是系统参数的微小变化会导致方程的解发生定性(即拓扑)变化的情况。分叉理论告诉我们，这种“因果”关系可以是高度非线性的，但通常只会以几种被称为分叉的一般方式出现，数学家们开始很好地理解这种方式。在目前的工作中，所研究的方程是非线性微分方程组；也就是确定系统如何随时间演变的方程。这类方程在科学中得到了广泛的应用，并有助于理解许多实际系统，例如在物理、工程、生物和气候方面。他说，这项研究计划有两个目标。在数学领域，通过在知识库中添加新类型的分支，将有助于分支理论本身的发展。在数学之外，这项研究计划将把分叉理论的思想应用到现实世界的系统中，目的是更好地了解这些系统可能发生的变化。从这一角度研究的一些系统的例子有：全球气候变化、机械系统的振动模式、人类某些心肺疾病的发病以及生态中的可持续生物多样性。在所有这些应用中，气候变化被选为本研究提案的优先事项。就在最近，我和同事们证实，地球气候系统可能会出现分叉，而且实际上在遥远的过去的气候中也发生过。一项初步分析得出的结论是，除非由二氧化碳产量增加导致的人为强迫得到严重遏制，否则未来我们将面临气候显著变暖的分叉。这一结果为政府间气候变化专门委员会发布的可怕预测增添了数学上的严谨性和严肃性。使用一系列更复杂的气候模型，建议对这种灾难性的分叉及其缓解进行进一步的研究。

In the study of systems of mathematical equations, Bifurcation Theory deals with cases in which small changes in the parameters of the system can lead to qualitative (i.e. topological) changes in the solutions of the equations. Bifurcation Theory tells us that this "cause and effect" relationship can be highly nonlinear, but typically will occur only in a few generic ways, called bifurcations, which mathematicians are beginning to understand very well. In the present work, the equations under study are systems of nonlinear differential equations; that is, equations that determine how a system evolves with time. Equations of this kind are widely used in science, and help in the understanding of many kinds of real systems, for example in physics, engineering, biology and climate.      This research program has two objectives. In the field of mathematics, it will contribute to the advancement of Bifurcation Theory itself, by adding new types of bifurcations to the knowledge base. Outside of mathematics, this research program will extend the application of ideas of Bifurcation Theory to real-world systems, with the goal of better understanding how these systems are likely to change. Some examples of systems that have been investigated from this perspective are: global climate change, patterns of vibrations in mechanical systems, onset of certain cardio-respiratory diseases in humans, and sustainable biodiversity in ecology.      Of all these applications, climate change is selected as the first priority for this research proposal. Very recently, coworkers and I have established that bifurcations may occur in the Earth's climate system, and actually have occurred in climates of the distant past. A preliminary analysis has concluded that a bifurcation to a dramatically warmer climate state awaits us in the future, unless anthropogenic forcing, caused by increased carbon dioxide production, is severely curtailed. This result adds mathematical rigour and gravitas to the dire predictions that have been issued by the Intergovernmental Panel on Climate Change. Further investigations of this catastrophic bifurcation and its mitigation are proposed, using a hierarchy of more sophisticated climate models.